Thresholded Covering Algorithms for Robust and Max-min Optimization

The general problem of robust optimization is this: one of several possible scenarios will appear tomorrow and require coverage, but things are more expensive tomorrow than they are today. What should you anticipatorily buy today, so that the worst-case covering cost (summed over both days) is minimized? We consider the k-robust model [6,15] where the possible scenarios tomorrow are given by all demand-subsets of size k. We present a simple and intuitive template for k-robust problems. This gives improved approximation algorithms for the k-robust Steiner tree and set cover problems, and the first approximation algorithms for k-robust Steiner forest, minimum-cut and multicut. As a by-product of our techniques, we also get approximation algorithms for k-max-min problems of the form: "given a covering problem instance, which k of the elements are costliest to cover?"